Probabilistic Extensions of the Erd\H os-Ko-Rado Property

The classical Erd\H os-Ko-Rado (EKR) Theorem states that if we choose a family of subsets, each of size (k), from a fixed set of size (n (n > 2k)), then the largest possible pairwise intersecting family has size (t ={n-1\choose k-1}). We consider the probability that a randomly selected family of size (t=t_n) has the EKR property (pairwise nonempty intersection) as $n$ and $k=k_n$ tend to infinity, the latter at a specific rate. As $t$ gets large, the EKR property is less likely to occur, while as $t$ gets smaller, the EKR property is satisfied with high probability. We derive the threshold value for $t$ using Janson's inequality. Using the Stein-Chen method we show that the distribution of $X_0$, defined as the number of disjoint pairs of subsets in our family, can be approximated by a Poisson distribution. We extend our results to yield similar conclusions for $X_i$, the number of pairs of subsets that overlap in exactly $i$ elements. Finally, we show that the joint distribution $(X_0, X_1, ..., X_b)$ can be approximated by a multidimensional Poisson vector with independent components.Comment: 18 page

Tracing Failure of Coral Reef Protection in Nonstate Market-Driven Governance

Institutional failure remains an important blind spot in the private governance literature. In this article we argue that a focus on scope conditions alone cannot explain why some programs thrive while others cease to exist. Studying the now-defunct Marine Aquarium Council—a certification program for coral reef protection—we adopt an institutional-process approach to fill this gap. Our main points can be summarized in a two-step argument: First, we argue that the scope conditions of private governance are partly endogenous to these processes. Through making strategic decisions, private governance programs have a certain level of control over their environment, and thus over the scope conditions under which they operate. Second, initial choices often unfold path dependencies over time. By tracing the evolution of the Marine Aquarium Council, we illustrate the program’s “mission creep” and the “vicious cycle” of self-reinforcing activity that culminated in its failure